Evaluating Piecewise and
Step Functions
Today, you will:
a) evaluate piecewise functions
b) investigate and explain characteristics of a variety of piecewise
functions including; domain, range, vertex, axis of symmetry,
extrema, points of discontinuity and rates of change
Evaluating Piecewise Functions
 Piecewise functions are functions defined by
at least two equations, each of which applies
to a different part of the domain
 There are several types of piecewise
functions that can take on all different shapes
and forms!
 A piecewise function looks like this:
Domain restrictions
Equations
Evaluating Piecewise Functions
 When we ‘evaluate’ piecewise functions, the most important
thing to do is look at the individual domains for the functions and
find which part of the piecewise function you will need to use.
For example, find
a. g(-2) and b. g(2)
At g(-2) we would use the top function
because -2 < 1. So, g(-2) = -2
At g(2) we would use the bottom function because
2 > 1. So, g(2) = 3(2) – 1 = 5
Evaluating Piecewise Functions
Lets look at another example.
Which equation would we use to find; g(-5)? g(-2)? g(1)?
g (5)  (5)  2(5)  1  16
2
g (2)  (2)  2(2)  1  1
2
g (1)  (1)  2(1)  1  4
2
Step Functions
Step functions are special types of piecewise functions that are defined
by a constant value over each part of its domain. Graphically, it looks
like a flight of stairs
An example of a step function:
Graphically, the equation would look like this:
Evaluating Step Functions
 To evaluate a step function, treat it just like any other
piecewise function. Using the domain, identify which
piece of the piecewise function you will need to use
and identify the value.
 Two special kinds of step functions are called “floor”
and “ceiling” functions. In ceiling functions, nonintegers are rounded up to the nearest integer. In
floor functions, all non-integers are rounded down.


Example: ceiling function – you use 1:47 talking on the
phone, but you are charged for 2 min.
Example: floor function – you may be 14 years and 8
months old, but you say you are 14 years old until your
15th birthday.
Characteristics of Piecewise Functions
 Piecewise functions, like all functions, have
special characteristics. Some are familiar,
some are new.
Domain and Range of Piecewise
Functions
 Domain (x): the set of all input numbers - will
not include points where the function(s) do
not exist. The domain also controls which part
of the piecewise function will be used over
certain values of x.
 Range (y): the set of all outputs.
Points of Discontinuity
 With piecewise functions, we have what are called points of
discontinuity. These are the points where the function either
“jumps” up or down or where the function has a “hole”.
 For example, in a previous example
Has a point of
discontinuity at 1
 The step function also has points of
discontinuity at
1, 2 and 3.
Axis of Symmetry
 In absolute value functions, there exists a
vertical line that splits the equation in half.
This ‘axis of symmetry’ can be found by
identifying the x-coordinate of the vertex (h,k),
so the equation for the axis of symmetry
would be x = h.
For the equation y  x 1 1 the
axis of symmetry is located at x = 1
Maxima and Minima
 Like all functions, piecewise functions have
maxima and minima. These values will be a
part of the range of the function
In this function, the
minimum is at y = 1 and the
maximum is infinity
In this function, the
minimum is at y = -2 and
the maximum is infinity
Intervals of Increase and Decrease
 By looking at the graph of a piecewise
function, we can also see where its slope is
increasing (interval of increase), where its
slope is decreasing (interval of decrease) and
where it is constant (slope is 0). We use the
domain to define the ‘interval’.
This function is decreasing on the
interval x < -2, is Increasing on the
interval -2 < x < 1, and constant
over x > 1